I would like to ask the following:
If an observer travels up or down in an elevator on earth, does he experience a horizontal force due to earth's rotation?
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$\begingroup$ What would be the reason of such force? $\endgroup$– WojciechMar 4, 2014 at 13:37
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$\begingroup$ The rotation of earth. I'll edit this. $\endgroup$– Mirc BreitschuhMar 4, 2014 at 13:46
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$\begingroup$ Congrats! You just discovered the Corriolis forces! $\endgroup$– John AlexiouMar 4, 2014 at 20:48
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2 Answers
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Surprisingly, the answer is that yes you do, though the effect is very small. To see this consider the following (highly exaggerated) diagram of the lift shaft:
The Earth rotates at a constant angular velocity of one rotation every 24 hours ($\omega = 7.27 \times 10^{-5}$ radians/sec). The tangential velocity of a part of the lift shaft at a distance $r$ from the centre of the Earth is $v_t = r\omega$ so the velocity $v_t$ increases with $r$. This means as you ascend the lift shaft you accelerate in a horizontal direction otherwise you'd be moving at a different speed to the lift.
We can easily calculate the force. Start with $v_t = r\omega$ and differentiate to get the tangential acceleration:
$$ a_t = \frac{dv_t}{dt} = \omega \frac{dr}{dt} $$
And $dr/dt$ is just the vertical speed (call this $u$). The force is just mass times acceleration, so the tangential force is:
$$ F_t = m \omega u $$
I don't know what speed lifts move at, but let's guess a 1 m/sec. My mass is about 70 kg, so when I'm going up in a lift the tangential force is:
$$ F_t = 70 \times 7.27 \times 10^{-5} \times 1 = 0.005N $$
answered Mar 4, 2014 at 14:29
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$\begingroup$ Thanks John. This is exactly the setup I had in mind as well. Would you also agree that this is basically what is refered to as the coriolis force, as explained by Dohn Joe (which seemes to make sense to me)? $\endgroup$ Mar 4, 2014 at 14:48
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$\begingroup$ @MircBreitschuh: we normally think of the Coriolis effect as bending the path of an object moving on the Earth's surface rather than normal to it. Still, I guess it is technically a Coriolis effect. $\endgroup$ Mar 4, 2014 at 16:22
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He may experience a Coriolis force, but that is very small in magnitude. I am not sure if you could measure it. The Coriolis force, however, is only experienced by observers in a moving coordinate system when moving relative to the moving frame of reference. As you situate your elevator on earth, we have a rotating coordinate system, that rotates with the earth.
